MAJOR P. A. MACMAHON ON THE THEORY OP PARTITIONS OF NUMBERS. 369 
n 
which reduces to 
1 — a&Xi. 1 — - X, 
a - 
1 - - X 3 .I - - X, 
h ' cd ' 
1 - XfX^X. 
1 - Xj. 1 - XjX.,. 1 - X 1 X 3 .1 - XjXoX,. 1 - XiX^XgX, 
establishing the ground solutions 
(a.,a3,a3, «4) = (1, 0,0,0); (1, 1, 0, O); (1,0, 1,0); (1, 1,1,0); (1, 1, 1, 1). 
connected by the syzygy indicated by 
X,. X1X2X3 - XiX,. X1X3 = 0, 
and leading to the enumerating function 
1 
(1 -x) (1 - x?f (1 - x=*)' 
Art. 83. If the parts be restricted not to exceed j in magnitude, we may take as 
n function 
1 - 
1 - o&Xi . 1 - — Xo 
1 - — X.. . 1 
b 
a 
cd, 
X, 
and herein putting Xi = X2 = X3 = X4 = a?, and reducing, we get 
1 - jl - x^^^ V- 1 - x^ +^ 
1 — X ' \ 1 — a;^ / ' 1 — ’ 
and we notice that we may represent this diagrammatically on the points of the 
original lattice, viz. :— 
( J+D 
(!) 
(J^-Z) 
(t) 
( J-¥2 ) ( J-¥3 > 
*2> (3) 
3 B 
VOL. CXCII.—A. 
